## Abstract

We compare the contextual probabilistic structures of the seminal two-slit experiment (quantum interference experiment), the system of three interacting bodies and *Escherichia coli* lactose–glucose metabolism. We show that they have the same non-Kolmogorov probabilistic structure resulting from multi-contextuality. There are plenty of statistical data with non-Kolmogorov features; in particular, the probabilistic behaviour of neither quantum nor biological systems can be described classically. Biological systems (even cells and proteins) are macroscopic systems and one may try to present a more detailed model of interactions in such systems that lead to quantum-like probabilistic behaviour. The system of interactions between three bodies is one of the simplest metaphoric examples for such interactions. By proceeding further in this way (by playing with *n*-body systems) we shall be able to find metaphoric mechanical models for complex bio-interactions, e.g. signalling between cells, leading to non-Kolmogorov probabilistic data.

## 1. Introduction

The two-slit experiment demonstrating the interference of probabilities lies at the heart of quantum foundations. We note that, for example, Feynman pointed to this experiment in practically all foundational discussions. In particular, he emphasized its non-classical probabilistic structure [1]. (It seems that Feynman was not aware of the modern measure-theoretic formulation of classical probability theory [2] and he referred to Laplace probability theory.) He highlighted the contextual structure of this experiment as the source of non-classicality and demonstrated by using the calculus of complex probability amplitudes (wave functions) that the basic law of classical probability theory, additivity of probability, is violated for data collected for three experimental contexts, *C*_{i}, *i*=1,2, only the *i*th slit is open, and *C*_{12}, both slits are open. This violation of additivity can be reformulated in terms of conditional probabilities [3] (which is more useful in the framework of probability inference) as violation of the *formula of total probability* (FTP).

Thus, the mathematical quantum formalism can be treated as one possible way to work with *non-Kolmogorov probability* (NKP). From this point of view, we can stress that the formalism of quantum mechanics is not just a physical theory but also a probability theory. We can apply NKP to non-physical phenomena which violate the additivity in classical probability theory. We can find many applications of quantum probability outside of physics, e.g. the application of quantum interference to psychological experiments [4,5] and the application of quantum mechanical dynamics to the Black–Scholes option pricing model in finance theory [6,7].

Recently, it was shown that many biological phenomena, such as metabolism, differentiation, evolution, adaptation and cognition, violate the FTP [3,8,9] (as well as another consequence of the classical Kolmogorov probability model—the Bell inequality [10], which plays a key role in modern debates on quantum foundations [11]; see [12] on the violation of the Bell-type inequalities in the process of recognition of ambiguous figures). As in the case of quantum physics, in biology the contextuality of behaviour is the source of the NKP structure of statistical experimental data. Bio-contextuality results from the *adaptivity* of bio-systems to the environment (in particular, to experimental contexts). By comparing the NKP features of the behaviour of bio-systems and quantum physical systems, one may try to apply mathematical quantum formalism to model probabilistic biological behaviour. We emphasize that the use of this formalism is just a possibility, not a necessity. Other NKP models may be used as well, e.g. *non-additive probabilities* [13]. In this study, we shall not discuss the advantages and disadvantages of the use of quantum probability (see [14,15] for a review). In [3,9,16–18], there are discussions on the availability of quantum mechanical formalism. Here, in this paper, we discuss only the following foundational problem. Bio-systems are macroscopic systems compared with the scales of quantum mechanics—even proteins and cells are huge compared with electrons and atoms. Therefore, in principle, their behaviour can be described by using classical physical and chemical interactions; for example, molecular biology studies of signalling in molecular networks inside a cell and neurophysiology in networks of firing neurons. Thus, although the probabilistic structures of quantum physical and quantum-like biological models are similar, functionally they are different, because in quantum mechanics we cannot generate probabilistic behaviour by using models with classical physical and chemical interactions—there are ‘no hidden variables’ (at least this is the common viewpoint; see, however, for example [19] for debates).

The aim of this article is to demonstrate that classical mechanical interactions inside sufficiently simple physical systems can generate NKP behaviour. It is easier to model mechanical interactions and derive from them the NKP output than to model interactions in even the simplest bio-systems, such as cells. Macroscopic biological processes involved in biological phenomena are usually deterministic, but they operate under a many complex interaction (information transduction) network [18,20]. The mathematical formulations of such macroscopic complex systems, including the complex network system of metabolism in a bacterium, require tremendous efforts (e.g. [21]).

Our aim is to determine that NKP behaviour can arise from a classical system accompanied by complicated interactions. In this sense, the three-body system can be a candidate for a metaphoric mechanical model of such NKP behaviour. The two-body problem, such as the typical Sun/Earth relation under gravity, can be mathematically described by the deterministic Newton's equation of motion. However, the three-body system is known not to be able to be solved analytically, although it is deterministic. This is because it contains more than two interactions. (It is surprising that the transition from two to three interactions results in a tremendous increase in the complexity of behaviour.)

In this study, we demonstrate the NKP features of the simplest physical system of three linearly connected balls, which is a special case of the three-body system. We try to apply our NKP theory to understand the fundamental property of the phenomenon observed in the three-body problem, but do not aim to solve the three-body behaviour itself. And this simple example will aid in understanding the fundamentals of our NKP applied to a complex system with many interacting components.

## 2. Probability structure of the two-slit experiment

Consider the following pair of observables *a* and *b*. We select *a* as the slit that is observable, i.e. *a* = 0,1, and observable *b* as the position on the photo-sensitive plate; set *P*(*i*)=*P*(*a*=*i*), *P*(*x*)=*P*(*b*=*x*). Then
2.1where *ψ*_{0} and *ψ*_{1} are two wave functions whose squared absolute values |*ψ*_{i}(*x*)|^{2} give the distributions of photons passing through the slit *i*=0,1. The term implies the interference effect of two wave functions. Set *P*(*x*|*i*)=|*ψ*_{i}(*x*)|^{2}, then equation (2.1) is represented as
2.2Here the values of probabilities *P*(0) and *P*(1) are equal to , since we consider the symmetric experimental setting of the source with respect to the slits. For the general experimental setting, *P*(0) and *P*(1) can be taken as the arbitrary non-negative values satisfying *P*(0)+*P*(1)=1. In the above form, the classical probability law (FTP)
2.3is violated, and the term of interference specifies the violation. The violation of the FTP is a consequence of the special contextual structure of the two-slit experiment (in fact, a group of experiments). As Feynman pointed out [1], the interference formula (2.2) involves three contexts: *C*_{i},*i*=1,2, only the *i*th split open, and *C*_{12}, both slits are open. They are represented by corresponding quantum states.

It is convenient to discretize all variables, i.e. select some domain on the photo-emulsion screen and set *b*=+, if a point appears in this domain, and *b*=−, in the opposite case. Thus, the state *ψ* representing context *C*_{01} can be expanded in two ways:
2.4where the state |*i*〉 represents context *C*_{i}, and
2.5

where the states |±〉 are eigenvectors of the (discrete) *b*-observable. Then Born's rule implies that
2.6and by using expression (2.4) of |*ψ*〉 we obtain the quantum version of the FTP with the additional (interference) term.

For simplicity, we assumed that *a* and *b* are represented as von Neumann–Lüders observables of the projection type and given by Hermitian operators. Thus 〈0|1〉=0,〈+|−〉=0. The real situation can be more complicated and, instead of observables of the projection type, generalized observables given by the so-called positive operator-valued measures (POVMs) have to be in use [22]. For POVMs, the interference coefficients
2.7can exceed 1, so they need not be represented in the form (see [19] or [22] for details). More generally, especially in non-physical applications, we can use the general theory of adaptive dynamics [18,20].

## 3. Lactose–glucose metabolism

A cell recognizes lactose (*L*) and glucose (*G*). In figure 1, we show the microbiological system of the metabolism of lactose and glucose. The response to the expression of the Lac operon as the adaptation to the chemical *L*/*G* context, *C*_{L/G}, is represented as − and +. (Roughly speaking, in a mixed lactose–glucose environment each lactose molecule contributes to a decrease in the level of metabolism.) In [23], we demonstrated (by using experimental statistical data) that functioning of the *Lac* operon leads to violation of the classical FTP:
3.1Probabilities *P*(*L*),*P*(*G*) are determined by the concentrations of *L*,*G* in the environment of a cell. The probabilities *P*(+),*P*(+|*L*),*P*(+|*G*) of activation of the *Lac* operon are determined by the following experimental contexts: *C*_{L/G}, mixed lactose–glucose environment; *C*_{L}, only lactose; *C*_{G}, only glucose. The destructive interference of *L* and *G* is represented as
3.2where λ<0 (see [21] for the concrete value of λ). Similar to (2.4) and (2.5), this multi-contextual setting is represented by the quantum formalism. Contexts *C*_{L/G},*C*_{L},*C*_{G} are expressed by the states |*ψ*〉,|*L*〉,|*G*〉. Then
3.3where the state |*i*〉 represents context *C*_{i}, and
3.4where |±〉 are the eigenvalues of the observable *Lac* operon activation. Finally, we again use Born's rule (2.6) to obtain the quantum-like FTP (3.2).^{1}

## 4. Non-Kolmogorov probabilistic features of the three-body system

In the previous section, we showed the quantum-like interpretation of *Escherichia coli* metabolism, and we interpreted the activation of β-galactosidase production as the consequence of the interaction of two sugars, glucose and lactose. Because massive amounts of genes and proteins are involved in metabolism in a cell, such interactions are complicated. Similarly, the behaviour of the three-body system is as a consequence of complicated interactions.

In figure 2, we can see that there is an analogous situation between the three-body system and the metabolism system in the cell. From the position of the sphere in the centre, we can obtain the area where the object exists. Then, we can divide the area into two parts with a certain boundary, e.g. *x*<*a* and *x*≥*a* with a boundary constant *a*. If the central sphere appears on one side, then we denote such an event by +, and if the central sphere appears on the other side, then we denote it by −. In this model, each ± can change into the other. This fractuation is similar to what happens in the cell of *E. coli* although we cannot see the frequent change between activation and non-activation of the Lac operon. Instead, we can measure the probability of + or − (see [21]).

Therefore, we can examine the non-Kolmogorov features of the three-body system. In the following sections, we use the three-body system to show the violation of the FTP.

### (a) Probabilistic interpretation of the complexity of the orbits

The system is represented in one dimension and is composed of three points with certain masses connected by two springs (see the model in figure 3). The masses and coordinates of particles 1, 2 and 3 are all *m*=1 and *x*_{1}, *x*_{2} and *x*_{3}, respectively. They are connected by two springs with spring constants, *k* and *k*′ (stiffnesses).

First, let us take *k*=10 and *k*′=0 in order to discuss the simple case of the two-body system. Then, the trajectory becomes the same as the analytical solution, i.e. the usual periodic vibration of a spring (figure 4*a*). Second, we take *k*=9.75, *k*′=0.25 for the three-body system. In addition, there is an analytical solution in this special case (see appendix A). The resultant trajectory of particle 1 is obtained as shown in figure 4*b*.

We define two quantities *p* and *q* to discuss the complexity of the one-dimensional three-body system.

The quantity *p* is given by the ratio of spring constants as
4.1Obviously, if *p*=0 or 1, then the trajectories of the two-body system appear. Thus, the parameter *p* encodes the information of the experimental situation (context) whether it is a two- or three-body system. This parameter reflects the basic features of the context. (Context is determined by the pair of parameters *k* and *k*′.) In our simulations, the values of parameters *k* and *k*′ will be fixed. And we shall simply speak about contexts *C*_{23}: both springs are present, and *C*_{i}: only the *i*th spring is present. These are the analogues of the corresponding contexts for the two-slit experiment: only one slit is open or both slits are open. These are also the analogues of the corresponding contexts for lactose–glucose metabolism in *E. coli* bacteria (figure 2).

Another quantity *q* is the probability of particle 1 being in range of *x*_{1}>*a*, where, just to be specific in the numerical simulation, we chose *a*=0.2,*a*=0.8. Formally, *q* is defined as
4.2Here, *n* is the number of time steps of particle 1 existing at *x*_{1}>*a*, and *N* is the number of time steps of the entire simulation.

Figure 5*a* shows the probabilities (*q*) of particle 1 being in a certain range of *x*_{1} within a certain time period *T*=40 under three-body interactions by changing the contributions of the two springs by *p*=*k*/(*k*+*k*′)(0≤*p*≤1), compared with the probabilities under of the respective independent two-body interactions (*p*=1 or *p*=0), which is solved analytically.

We can practically repeat the discussion on lactose–glucose interference. We can say that particle 1 ‘recognizes’ particles 2, 3 (as a bacterium recognizes molecules of lactose and glucose). The ‘response’ of 1 is + if it is in the range *x*_{1}>*a* (*a*=0.2,*a*=0.8 in our numerical simulation) and − in the opposite case. Our numerical simulation demonstrated (figure 5*b*) that the interaction of particle 1 and particles 2, 3 leads to violation of the classical FTP:
4.3Probabilities *P*(2),*P*(3) are determined by the contributions of the springs: *P*(2)=*p*,*P*(3)=1−*p*. Probabilities *P*(+),*P*(+|2),*P*(+|3) of particle 1 being in the range *x*_{1}>*a* (the *q* parameter in the above consideration) are determined by experimental contexts *C*_{23},*C*_{2},*C*_{3}. Interference of the interactions of particle 1 with particles 2,3 is represented as the appearance of the interference term perturbing the classical FTP:
4.4The results of the numerical simulation for λ are presented in §4b.

Similar to the two-slit experiment (2.4), (2.5) and the Lac operon activation experiment (3.3), (3.4), we can represent this multi-contextual setting by using the quantum formalism. Contexts *C*_{23},*C*_{2},*C*_{3} are represented by the states |*ψ*〉,|2〉,|3〉. Then
4.5where the state |*i*〉 represents context *C*_{i},*i*=2,3, and
4.6where |±〉 are eigenvalues of the ‘being in the range *x*_{1}>*a* observable.’ Finally, we once again use Born's rule (2.6) to obtain the quantum-like FTP (4.4).

### (b) Quantum-like interference coefficient for interactions in three-body systems

By numerical simulation, we can estimate the values of conditional probabilities: 4.7

We change the value of the spring constants as follows:
4.8Then we have the probabilities (*p*_{i},*q*_{i}) for each pair of spring constants. We can obtain the value of λ_{i} in equation (3.2) by using numerical simulation (figure 6).

We show the value of λ for the range of {*x*>0.8}, and λ for the range of {*x*>0.2}. Here, we fixed the *k*+*k*′=10. As we can see, the interference coefficient λ takes a negative value. Thus, the interference is destructive. This coefficient exceeds −1, when *p* is around zero or 1. Figure 6 shows the plot with respect to *p* in [0.05,0.95]. There exists the λ whose value exceeds −1. Thus, the quantum-like representation (4.5), (4.6) is idealized and one has to proceed with POVMs or in the framework of general adaptive dynamics [24].

Also, for *p* in the middle range (around 0.5), the intensity of interference seems to be very weak, since λ takes values around zero. We interpret the case *k*=*k*′ as the classical mixture.

## 5. Conclusion

We identified the interference term for the three-body system, the simplest complex system. We proceeded in the same way as in quantum physics (the two-slit experiment [1]) and quantum-like modelling in molecular biology (*E. coli* diauxic growth [23]). The interference term depends on the spring's stiffness. The results show that the classical three-body system which has more than two interactions at the same time on at least one member of the system can exhibit non-classical probabilistic features, thus it can be treated as the NKP system. Again, as well as in the case of the two-slit experiment and *E. coli* diauxic growth, violation of the FTP can be modelled by using the Hilbert state formalism of quantum theory: state superposition and Born's rule.

This classical three-body system, which provides a metaphor for quantum probabilistic behaviour, is important to justify the use of the quantum formalism in biology, where stochasticity is a consequence of functioning of a complex network of interactions between molecules and cells. Now it is clear that classical interactions between subsystems of a complex system can lead to non-classical (quantum-like) probabilistic features. This should stimulate further applications of quantum-like methods in biology.

## Authors' contributions

M.A. designed the study and analysed the data; A.K. wrote the manuscript and participated in data analysis; M.O. coordinated the study and helped in the design/analysis of the study and manuscript preparation; Y.T. wrote the manuscript, participated in the design of the study and participated in data analysis; I.Y. carried out the simulation study, designed the study and analysed the data.

## Competing interests

We declare we have no competing interests.

## Funding

A.K. was partially supported by a grant from the programme Mathematical Modeling of Complex Hierarchic Systems of the Faculty of Technology, Linnaeus University, and a QBIC-grant from the Tokyo University of Science.

## Appendix A. Analytical solution of the one-dimensional three-body problem

Here, we use the notation shown in §4a. Newton's equation of motion is written in matrix form:
A 1
A 2Here, we rewrite the above equation with the new coordinates *y*_{1}=*x*_{1}, *y*_{2}=*x*_{2}+5 and *y*_{3}=*x*_{3}−5 as follows:
A 3
A 4The 3×3 matrix in equation (A 4) is diagonalizable if *k*≠*k*′. Therefore, we can rewrite equation (A 4) as
A 5Here, *α*_{i} is an eigenvalue of the matrix. Let (*z*_{1},*z*_{2},*z*_{3})^{t} be the *A*(*y*_{1},*y*_{2},*y*_{3})^{t}, then we have simple equations:
A 6Therefore, we can obtain solutions of with constants *Z*_{i}, then the solution *y*_{i} as the linear combination of *z*_{j}:
A 7Thus, we can have the analytical solution of *x*_{i} in the settings of figure 3. The eigenvalues *α*_{i} determine the periodicity of the orbits. However, the orbit is not periodic, but chaotic in general.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum foundations: information approach’.

↵1 We again simplified the real situation by working with observables of the projection type. As has already been pointed out, POVMs and general adaptive dynamics can represent interference coefficients exceeding 1. And this was the case in our study [21] based on the statistical data.

- Accepted February 15, 2016.

- © 2016 The Author(s)